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Properties

Label 475.2.e.d
Level $475$
Weight $2$
Character orbit 475.e
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(26,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3518667.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 3) q^{4}+ \cdots + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} - \beta_{3} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots - 3) q^{4}+ \cdots + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + q^{3} - 7 q^{4} + 6 q^{6} - 4 q^{7} + 12 q^{8} - 4 q^{9} - 10 q^{11} + 8 q^{12} - 15 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} - 28 q^{18} + 12 q^{21} - 8 q^{22} + 4 q^{23} + 21 q^{24} - 10 q^{26} + 16 q^{27} + 11 q^{28} + 2 q^{29} - 2 q^{31} - 6 q^{32} + 11 q^{33} + 25 q^{34} - 3 q^{36} + 4 q^{37} + 19 q^{38} - 10 q^{39} + 2 q^{41} - 23 q^{42} - q^{43} + 18 q^{44} - 48 q^{46} + 6 q^{47} + q^{48} - 14 q^{49} + 6 q^{51} - 35 q^{52} + 11 q^{53} - 10 q^{54} + 30 q^{56} + 19 q^{57} + 14 q^{58} - 6 q^{59} + 9 q^{61} - 13 q^{62} + 9 q^{63} - 16 q^{64} - 29 q^{66} - 20 q^{67} - 68 q^{68} - 10 q^{69} + 29 q^{71} + 11 q^{72} - 22 q^{73} + 7 q^{74} - 19 q^{76} + 32 q^{77} + 30 q^{78} + 24 q^{79} + 21 q^{81} + 31 q^{82} + 6 q^{83} - 56 q^{84} + 32 q^{86} - 24 q^{87} + 18 q^{88} + 14 q^{89} + 10 q^{91} + 41 q^{92} + 6 q^{93} + 42 q^{94} + 34 q^{96} + 7 q^{97} + 23 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 21\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.25351 − 2.17114i
0.610938 + 1.05818i
1.14257 + 1.97899i
−1.25351 + 2.17114i
0.610938 − 1.05818i
1.14257 − 1.97899i
−1.25351 − 2.17114i 0.610938 + 1.05818i −2.14257 + 3.71104i 0 1.53163 − 2.65287i 0.221876 5.72889 0.753509 − 1.30512i 0
26.2 0.610938 + 1.05818i 1.14257 + 1.97899i 0.253509 − 0.439091i 0 −1.39608 + 2.41808i 1.28514 3.06327 −1.11094 + 1.92420i 0
26.3 1.14257 + 1.97899i −1.25351 − 2.17114i −1.61094 + 2.79023i 0 2.86445 − 4.96137i −3.50702 −2.79216 −1.64257 + 2.84502i 0
201.1 −1.25351 + 2.17114i 0.610938 − 1.05818i −2.14257 − 3.71104i 0 1.53163 + 2.65287i 0.221876 5.72889 0.753509 + 1.30512i 0
201.2 0.610938 − 1.05818i 1.14257 − 1.97899i 0.253509 + 0.439091i 0 −1.39608 − 2.41808i 1.28514 3.06327 −1.11094 − 1.92420i 0
201.3 1.14257 − 1.97899i −1.25351 + 2.17114i −1.61094 − 2.79023i 0 2.86445 + 4.96137i −3.50702 −2.79216 −1.64257 − 2.84502i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.d 6
5.b even 2 1 95.2.e.b ✓ 6
5.c odd 4 2 475.2.j.b 12
15.d odd 2 1 855.2.k.g 6
19.c even 3 1 inner 475.2.e.d 6
19.c even 3 1 9025.2.a.z 3
19.d odd 6 1 9025.2.a.ba 3
20.d odd 2 1 1520.2.q.j 6
95.h odd 6 1 1805.2.a.g 3
95.i even 6 1 95.2.e.b ✓ 6
95.i even 6 1 1805.2.a.h 3
95.m odd 12 2 475.2.j.b 12
285.n odd 6 1 855.2.k.g 6
380.p odd 6 1 1520.2.q.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b ✓ 6 5.b even 2 1
95.2.e.b ✓ 6 95.i even 6 1
475.2.e.d 6 1.a even 1 1 trivial
475.2.e.d 6 19.c even 3 1 inner
475.2.j.b 12 5.c odd 4 2
475.2.j.b 12 95.m odd 12 2
855.2.k.g 6 15.d odd 2 1
855.2.k.g 6 285.n odd 6 1
1520.2.q.j 6 20.d odd 2 1
1520.2.q.j 6 380.p odd 6 1
1805.2.a.g 3 95.h odd 6 1
1805.2.a.h 3 95.i even 6 1
9025.2.a.z 3 19.c even 3 1
9025.2.a.ba 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 7T_{2}^{4} - 8T_{2}^{3} + 43T_{2}^{2} - 42T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 5 T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} + 2 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 25)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{5} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} + 133T^{3} + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 6 T - 7)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} + \cdots + 227)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + \cdots + 1369 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( T^{6} - 11 T^{5} + \cdots + 96721 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$67$ \( T^{6} + 20 T^{5} + \cdots + 7744 \) Copy content Toggle raw display
$71$ \( T^{6} - 29 T^{5} + \cdots + 218089 \) Copy content Toggle raw display
$73$ \( T^{6} + 22 T^{5} + \cdots + 5929 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + \cdots + 61504 \) Copy content Toggle raw display
$83$ \( (T^{3} - 3 T^{2} - 54 T - 77)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( T^{6} - 7 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
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